To show that X indeed measures the sensitivity of Z to changes in the constraint, let us perform a comparative-static analysis on the first-order condition (12.8). Since A, x, and y are endogenous, the only available exogenous variable is the constraint parameter c. A change in c would cause a shift of the constraint curve in the xy plane and thereby alter the optimal solution. In particular, the effect of an increase in c (a larger budget, or a larger production quota) would indicate how the optimal solution is affected by a relaxation of the constraint.
To do the comparative-static analysis, we again resort to the implicit-function theorem. Taking the three equations in (12.8) to be in the form of FJ(A, x, y; c) = 0 (with j = 1,2, 3), and assuming them to have continuous partial derivatives, we must first check that the following endogenous-variable Jacobian (where fxv =fvx> and Sxv =
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